List of top Linear Algebra Questions

Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?

For \(g\isin\Z\), let \(\bar{g}\isin\Z_{37}\) denote the residue class of g modulo 37. Consider the group U₃₇ = {\(\bar{g}\isin\Z_{37}:1\le g\le37\) with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?

The number of distinct subgroup of Z₉₉₉ is_____.

The number of elements of order 12 in the symmetric group S₇ is equal to____

Let σ be the permutation in the symmetric group S₅ given by
σ(1) = 2, σ(2) = 3, σ(3) = 1, σ(4) = 5, σ(5) = 4.
Define
N(σ) = {𝜏 ∈ S₅ ∶ σ ∘ 𝜏 = 𝜏 ∘ σ}.
Then the number of elements in N(σ) is equal to _________.

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?

Let be a finite group of order at least two and let e denote the identity element of G. Let σ: G →g be a bijective group homomorphism that satisfies the following two conditions:
(i): if σ(g)=g for some gεG, then g=e, 
(ii) (σ o σ) (g)=g for all g σ G. 
The n which of the following is/are correct ?

No. of group homomorphism from Z₄ to S₄